The Workshop is organized by the Department of Probability Theory, Statistics and Actuarial Mathematics of Taras Shevchenko National University of Kyiv

All sessions are organized as Zoom meetings by following the common link

Topic: International Workshop "Random Fields and Their Applications"

Time: Apr 14, 2022 11:55 AM Kyiv

Time is everywhere local time in Kyiv (GMT+3)

**Program committee:** Nikolai Leonenko, Yuliya Mishura, Anatoliy Malyarenko, Andriy Olenko

**Organizing committee:** Iryna Bodnarchuk, Kostiantyn Ralchenko

#### Invited Speakers

**Antoine Ayache**, University of Lille, France

Title of the talk: *On the monofractality of many stationary continuous Gaussian fields*

In this talk we focus on a general real-valued continuous stationary Gaussian field X characterized by its spectral density |g|^{2}, where g is any even real-valued deterministic square integrable function. Our starting point consists in drawing a close connection between critical Besov regularity of the inverse Fourier transform of g and α_{X} the random pointwise Hölder exponent function of X, which measures local roughness/smoothness of its sample paths at each point. Then, thanks to Littlewood-Paley methods and Hausdorff-Young inequalities, under weak conditions on g, we show that the random function αX is actually a deterministic constant which does not change from point to point. This result means that the field X is of monofractal nature. Also, it is worth mentioning that such a result can easily be extended to the case where X is no longer stationary but has stationary increments.

**Alexander Ivanov**, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Ukraine

Title of the talk:
*LSE Estimator Asymptotic Normality of the Symmetric Textured Surface Parameters*

A multivariate trigonometric regression model is considered. Various
discrete modifications of the similar bivariate model received serious attention
in the literature on signal and image processing due to multiple applications in
the analysis of symmetric textured surfaces. In the presentation asymptotic
normality of the least squares estimator for amplitudes and angular frequencies
is discussed in multivariate trigonometric model assuming that the random noise
is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly
dependent random field on R^M, M>2.

**Nikolai Leonenko**, Cardiff University, UK

Title of the talk:
*Sojourn functionals for spatiotemporal Gaussian random fields with long–memory*

This paper [1] addresses the asymptotic analysis of sojourn functionals
of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time
also known as long memory. Specifically, reduction theorems are derived for local
functionals of nonlinear transformation of such fields, with Hermite rank m≥1,
under general covariance structures. These results are proven to hold, in particular,
for a family of non–separable covariance structures belonging to Gneiting class.
For m=2, under separability of the spatiotemporal covariance function in space
and time, the properly normalized Minkowski functional, involving the modulus of
a Gaussian random field, converges in distribution to the Rosenblatt type limiting
distribution for a suitable range of the long memory parameter. For spatiotemporal
isotropic stationary fields on sphere similar results obtained in Marinucci
et al. [3]. Some other related results can be found in Makogin and Spodarev [2].

This is joint results with M.D.Ruiz-Medina (Granada University, Spain).

References

[1] Leonenko, N.N. and Ruiz-Medina, M.D. (2022) Sojourn functionals for spatiotemporal Gaussian random fields with Long-memory, Journal of Applied Probability, in press.

[2] Makogin, V. and Spodarev, E. (2022). Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence. Stochastics 94, 111–142.

[3] Marinucci, D., Rossi, M. and Vidotto, A. (2020). Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2 × R. Annals of Applied Probability, 31, 2311–2349

**Anatoliy Malyarenko**, Mälardalen University, Sweden

Title of the talk:
*Random fields forever! New applications for a classical theory* (Jointly with M. Ostoja-Starzewski)

The notion of a scalar-valued homogeneous and/or isotropic random field
on a Euclidean space or a sphere studied in the classical book by M.I. Yadrenko
"Spectral theory of random fields", can be generalized to the vector and
tensor-valued case in a nontrivial way. Recently such fields found interesting
applications in Continuum Physics and Cosmology. In this presentation, we give
a short description of some recent results obtained by the authors in this rapidly
growing area.

**Domenico Marinucci**, University of Rome "Tor Vergata", Italy

Title of the talk:
*Fluctuations of Level Curves for Sphere-cross-Time Random Fields*

The investigation of the behaviour for geometric functionals
of random fields on manifolds has drawn recently considerable
attention. In this paper, we extend this framework by considering
fluctuations over time for the level curves of general isotropic
Gaussian spherical random fields. We focus on both long and short
memory assumptions; in the former case, we show that the fluctuations of
u-level curves are dominated by a single component, corresponding to a
second-order chaos evaluated on a subset of the multipole components for
the random field. We prove the existence of cancellation points where
the variance is asymptotically of smaller order; these points do not
include the nodal case u = 0, in marked contrast with recent results on
the high-frequency behaviour of nodal lines for random eigenfunctions
with no temporal dependence. In the short memory case, we show that all
chaoses contribute in the limit, no cancellation occurs and a central
limit theorem can be established by Fourth-Moment Theorems and a
Breuer-Major argument.

Based on a joint paper with Maurizia Rossi and Anna Vidotto.

**Andriy Olenko**, La Trobe University, Australia

Title of the talk:
*Limit theorems for multifractal products of random fields* (Jointly with I. Donhauzer)

We will discuss asymptotic properties of multifractal products of
random fields. The obtained limit theorems provide sufficient conditions for
the convergence of cumulative fields in the spaces L_q. New results on the
rate of convergence of cumulative fields will be presented. Simple unified
conditions for the limit theorems and the calculation of the Renyi function
are given. They are less restrictive than those in the known one-dimensional
results. The developed methodology is also applied to multidimensional
multifractal measures. Finally, a new class of examples of geometric
sub-Gaussian random fields will be presented. In this case, the general
assumptions have a simple form and can be expressed in terms of covariance
functions only.

The talk is based on the joint results with I. Donhauzer in the manuscript.

**Giovanni Peccati**, University of Luxembourg, Luxembourg

Title of the talk:
*Functional convergence of nodal random fields*

We consider the random point process given by the nodal intersections
of planar Gaussian random waves and study its scaling limit over growing domains -
with specific emphasis on the exact variance asymptotics and functional convergence
of its chaotic components. One intriguing result is that the second chaotic
component converges to a total disorder field (that is, to a random field containing
uncountably many independent standard Gaussian random variables) indexed by closed
curves. Such a field already appears in works by Lebowitz (1983) (Coulomb systems)
and Buckley and Sodin (2017) (zeros of Gaussian entire functions).

Joint work with M. Notarnicola and A. Vidotto.

**Emilio Porcu**, Khalifa University, UAE

Title of the talk: *Regularity properties of Gaussian fields evolving temporally over spheres and manifolds*

**Dmitrii Silvestrov**, Stockholm University, Sweden

Title of the talk:
*Limit Theorems for Randomly Stopped Random Fields*

The purpose of this lecture is to present general conditions of
convergence in distribution for randomly stopped random fields, defined and taking
values in metric spaces, and to clarify the connection of these theorems with limit
theorems for randomly stopped stochastic processes. The corresponding results and
the bibliography of works in the area are presented in [1, 2].

References

[1] Silvestrov, D.S. (2004). Limit Theorems for Randomly Stopped Stochastic Processes.
Probability and Its Applications, Springer, London, xvi+398 pp.

[2] Silvestrov, D.S. (2021). Convergence in distribution for randomly stopped random
fields. Theor. Probab. Math. Statist., 105, 137-149.

**Evgeny Spodarev**, Universität Ulm: Institut für Stochastik, Germany

Title of the talk:
*Extrapolation of random fields via level sets* (Jointly with V. Makogin and A. Das)

Kriging methods are classically used for the extrapolation of square
integrable random fields. For heavy tailed random processes and fields, this theory
cannot be applied.
We fill this gap and construct a unified approach for the extrapolation of stationary
(possibly heavy tailed) random functions using the comparison of their level sets.
For that, a new excursion metric for random variables is defined, and its properties
are studied. Then it is shown how the new metric is connected to the error-in-measure
of level sets of the process and it predictor. For the case of Gaussian processes and
linear prediction, the new metric is minimised subject to the constraint that the
predictor has the same marginal distribution as the original process. Properties of
the new prediction such as consistency and exactness are studied. In the Gaussian case,
the existence and uniqueness of the solution of the corresponding linear programming
problem with quadratic constraints are analysed. Numerical examples round up the talk.

**Michele Stecconi**, Nantes University, France

Title of the talk:
*Geometry of Gaussian isotropic spin random fields*

I will talk about Gaussian isotropic spin random fields on the Riemann
sphere, i.e random sections of complex line bundles (the spin weight is
half of the degree of the bundle). I will consider those models that
exhibit a scaling limit when a parameter grows to infinity.
This setting includes sequences of Gaussian pure spin-weighted
harmonics, with different asymptotic regimes of the spin and the
eigenvalue, that yield different complex fields on the plane as their
scaling limit: the Berry (when the spin grows more slowly than the
eigenvalue) and the Bargmann-Fock (when the spin grows as fast as the
eigenvalue) models, in particular.

I will discuss how the scaling limit determines the leading term of
the asymptotic expected geometry in the form of expected Betti numbers
and Lipschitz-Killing curvatures of a class of singular loci of such
fields including: zeroes, excursion sets, critical points of the norm,
and semialgebraic singularities in general.

In particular, the asymptotic of the above quantities is the same for
all sequences of Gaussian spin-weighted harmonics for which the spin
weight grows more slowly than the eigenvalue.

**Yimin Xiao**, Michigan State University, USA

Title of the talk:
*Regularity and Geometric Properties of Gaussian Random Fields*

We present some results on regularity and geometric properties of Gaussian random fields. In particular, we prove four types of limit theorems: the
law of the iterated logarithm, uniform modulus of continuity, Chung’s law
of the iterated logarithm, and the modulus of nondifferentiability.

For this purpose, we formulate a framework that will be convenient for
studying the solutions of stochastic partial differential equations. An important condition in this framework is the property of strong local nondeterminism.